L(s) = 1 | + (1.54 + 4.75i)5-s + 0.206·7-s − 15.0i·11-s + 11.6i·13-s − 18.1i·17-s − 19.3i·19-s − 27.2·23-s + (−20.2 + 14.7i)25-s − 44.4·29-s + 20.3i·31-s + (0.319 + 0.982i)35-s + 18.1i·37-s + 32.3·41-s + 4.06·43-s + 5.37·47-s + ⋯ |
L(s) = 1 | + (0.309 + 0.950i)5-s + 0.0295·7-s − 1.36i·11-s + 0.899i·13-s − 1.07i·17-s − 1.02i·19-s − 1.18·23-s + (−0.808 + 0.588i)25-s − 1.53·29-s + 0.657i·31-s + (0.00913 + 0.0280i)35-s + 0.489i·37-s + 0.787·41-s + 0.0944·43-s + 0.114·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8004350312\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8004350312\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.54 - 4.75i)T \) |
good | 7 | \( 1 - 0.206T + 49T^{2} \) |
| 11 | \( 1 + 15.0iT - 121T^{2} \) |
| 13 | \( 1 - 11.6iT - 169T^{2} \) |
| 17 | \( 1 + 18.1iT - 289T^{2} \) |
| 19 | \( 1 + 19.3iT - 361T^{2} \) |
| 23 | \( 1 + 27.2T + 529T^{2} \) |
| 29 | \( 1 + 44.4T + 841T^{2} \) |
| 31 | \( 1 - 20.3iT - 961T^{2} \) |
| 37 | \( 1 - 18.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 32.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 4.06T + 1.84e3T^{2} \) |
| 47 | \( 1 - 5.37T + 2.20e3T^{2} \) |
| 53 | \( 1 + 79.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 83.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 36.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.51T + 4.48e3T^{2} \) |
| 71 | \( 1 + 41.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 41.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 15.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 50.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 10.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 12.1iT - 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.242819048113952835320901114250, −8.274342923108272886956024324322, −7.35548974971844648455441896580, −6.61472967291891458656711664213, −5.92164307450560261954588716026, −4.95278147364991328555022253694, −3.73364826338102481153247477045, −2.93277066136664978209034109941, −1.88305036305111290972540833448, −0.21218665288960495769378857195,
1.40677080180912845535223061696, 2.24114717593397093011791419016, 3.84148527231534127864915732242, 4.45780959602155916111940336104, 5.64065849248779684451988476093, 6.00333788851436005818152240984, 7.45541630481671915353801088422, 7.928008537323925449646710441746, 8.809639165511330440943479428912, 9.735637337128141397331994458813